Fractal analysis has already found widespread application
in the field of neuroscience and is being used in many other
areas. Applications are many and include ion channel kinetics
of biological membranes and classification of neurons according to their branching characteristics. In this article we
review some practical methods that are now available to allow
the determination of the complexity and scaling relationships
of anatomical and physiological patterns. The problems of describing fractal dimensions are discussed and the concept of
fractal dimensionality is introduced. Several related methodological considerations, such as preparation of the image and
estimation of the fractal dimensions from the data points,
as well as the advantages and problems of fractal geometric
analysis, are discussed. 䉷 2001 Academic Press

One of the basic tenets of neurobiology holds that
the function of a nerve cell is largely dependent on
its structure. To understand how a neuron integrates
its myriad synaptic inputs to generate an appropriate response, a thorough understanding of the
cell’s morphology and geometry is required. Thus
many quantitative parameters have been used to
characterize the morphology of nerve cells. The simple models that have been used so far for shapes,
such as spheres, ellipsoids, and polyhedra, and their
corresponding two-dimensional profiles, are useful
for many purposes, including estimates of volume or
size distribution, but certainly fall short in dealing
1
To whom all correspondence and reprint requests should be
addressed at School of Community Health, Charles Sturt University, Albury, 2640, N.S.W., Australia. Fax: ⫹61 2 516772. E-mail:
hjelinek@csu.edu.au.

1046-2023/01 $35.00
Copyright 䉷 2001 by Academic Press
All rights of reproduction in any form reserved.

with neuronal complexity. One method for describing
the irregular shape of feature profiles has been to
“unroll” that shape by plotting distance from the
centroid as a function of angle, and then to perform
a Fourier analysis on the resulting curve (1). Besides
being computationally demanding, this approach has
difficulty in dealing with shapes so irregular that
the radius line may intersect the outline more than
once. Other approaches such as the form factor are
also not useful since it can be the same for different
shapes and is not able for instance to measure quantitatively the complexity or degree of dendritic
branching (1). Sholl analysis has been widely used
to analyze branching characteristics of neurons. This
method determines the number of intersections of
dendrites with a set of concentric circles (2). Sholl
analysis is not an ideal method to measure neuronal
complexity as it has several problems with determining the number of processes at each level of measurement. Furthermore it is insensitive to the border
ruggedness of neurons (3, 4).
A new approach to this problem requires us to
think about the concepts of fractal geometry. In the
short period since Mandelbrot defined the term fractal (5), the field of fractal geometry has enjoyed an
enormous surge of popularity. The key observation
is that structures growing according to stochastic
processes are not really as disordered as they seem at
first glance. A nontrivial, scale invariant symmetry
over several orders of magnitude has been found to be
a typical principle of order for such growth processes,
which can be quantified by the fractal dimension.
Although this parameter can be used to quantify the
309

This
addition of detail results in an ideal fractal object
having an infinite boundary length (16.
THEORETICAL CONSIDERATIONS OF
FRACTAL GEOMETRY AND NATURALLY
OCCURRING FRACTALS
An object is said to be fractal if certain criteria such
as the object being self-similar or scale invariant are
met.310
´ NDEZ AND JELINEK
FERNA
complexity of the borders of a neuron (6–12) and to
measure how completely the branches of a neuron
fill its dendritic field (13. Figure 1 shows an approximation of an ideal/
theoretical fractal with a fractal dimension of 1.
FIG. Construction of the Koch curve with a D of 1. Thus natural patterns display statistical self-similarity only between
an upper and lower bound. that is. Computer-generated fractals. It is called dimension because it provides a measure of how completely
an object fills space. The form
of this object is complex since any change in
magnification/scale will show more detail to the resolution limit as the magnification is increased.
it is equal to the standard Euclidean dimension for
which an ideal point has a dimension of 0.
FRACTAL DIMENSIONS
An important parameter in fractal analysis of biological structures is the fractional or fractal dimension (D). We also review some of the methodologies available for calculating the fractal dimension. Limitations are also imposed by recording and imaging techniques. they do not have a characteristic
unit of length. such as the Koch curve. and so on) the fine
detail of the complex curve would be lost due to the resolution
limits of the printing process. and a perfectly solid volume has a dimension of 3. Thus fractals are always described by power
functions since homogeneous power laws lack natural scales. the advantages and problems of fractal geometry. E. The fractal dimension for this purpose is
therefore not intended to indicate whether the image
is a fractal object. Raising equilateral triangles from the middle third
of each of the line segments in the object produces the image in
(C). The final
value of the amount of detail or irregularity at different scales associated with a natural object can then
be determined by the use of fractal analysis.
and does not necessarily imply any biological process
nor mechanism involved in their development. 17). When D takes an integer value. Then the middle third is raised to produce an equilateral
triangle (B). time. Mandelbrot has shown that the boundary length of a fractal
object can be mathematically expressed as a power
law.26
that was described by the Swedish mathematician. Further it should be kept
in mind that all natural objects are.
Helge von Koch in 1904. Many patterns in
biology display a limited self-similarity or approximate self-similarity. unavoidably finite and limited in scale by their own nature. in contrast with
mathematical fractals. and some of its current applications in
neuroscience. are sometimes termed
prefractals since they are limited resolution images
and therefore do not realize the detail implicit in the
complete mathematical formulation (15).26.
. D is called fractal
because it usually is not an integer. At higher stages of construction (D. The
sequential construction of this fractal begins with a straight line
(A). an ideal
line has a dimension of 1. 14) it should be noted that
the fractal dimension is only a descriptive parameter. They are generally held to be
statistically self-similar. or mass (16). which increases in value with increasing
structural complexity and describes the “fractured”
nature of objects in nature (10). 1.
like the dendritic field area or the size of the soma.
In this paper we emphasize that fractal analysis
is a useful tool for improving image description and
for categorizing images representing morphologically complex objects based on the value of the fractal
dimension. an ideal plane has a dimension of 2.

For example. Hausdorff (32) suggested that the volume or measure of the sphere
should be eD where e equals the resolution of measurement. drawn from throughout the retina. 2.
The beta cell (on the right) has a more profuse branching pattern than the gamma cell (on the left) or the alpha cell (in the center). 3). It is calculated by covering an object with countable spheres whose radii are not greater than the
image but decrease to zero. nerve cells seen in two dimensions are not straight lines. 19).
It is not easy to give a precise definition of a fractal
(15). with their associated box counting dimensions. the volume
goes either to zero or infinity. 1. and there are in the literature many different
types of fractal dimensions so that even research
mathematicians are not agreed on their names or
equivalence (18. Examples of different cat ganglion cells. a very good estimate of D can be achieved
by different fractal analysis methods (Fig. neurons with
low D values.45 (Fig. Since many of these fractal dimensions
are used mainly in pure mathematics or applied
physics. A straight line is drawn from the
cell silhouette to its value on the D axis.
Calculating the Hausdorff dimension is generally
FIG. for instance.2. Table 1 lists some of the most
important fractal dimensions with their synonyms
and context.
Hausdorff Dimension
The original intention of Hausdorff was to define
a parameter that was independent of the resolution
of measurement and was applicable to all shapes
(16). All
methods rely on the relationship between a measuring device and the object’s spatial distribution. Various other aspects of fractal analysis and D are discussed formally by other
authors (15. would have relatively few dendritic branches and cover the two-dimensional area
less completely than neurons with higher D values
like 1. or other experimental data obtained from presentations of natural objects.USE OF FRACTAL THEORY IN NEUROSCIENCE
Since. say 1. This definition of dimension was extended and put into a more systematic framework
by Besicovitch (33). 2). 20–31). this would be when D ⫽ log
4/log 3 ⫽ 1.26.
. The
fractal dimensions are sufficiently different to suggest that they represent distinct ganglion cell types. 16. their D values fall between 1 and 2.
METHODS FOR DETERMINING FRACTAL
DIMENSIONS
Although the mathematically rigorous determination of D is impossible for a fractal point set obtained
311
from digitized photographs. and they do not completely cover the two-dimensional area. The D-dimensional Hausdorff measure of
an image is finite only when D (the dimension value)
equals the dimension of the image. 18. Measuring any self-similar set with spheres of integer dimension. drawings. For the Koch
curve shown in Fig. we consider only those that are potentially
useful in neuroscience.

.50. After
dilation with a disk kernel diameter of 16 pixels. and Weibel.
usually is greater than or
equal to the Hausdorff
dimension
Often used in calculating the
fractal dimension of outlines
Used for calculating the fractal
dimensions of many biological
structures in 2D and 3D
Used in the context of clusters and
networks. D(2) in
multifractal analysis
Context
Generic term for fractal dimension
Widely used in pure mathematics.. a
ruler of decreasing size r is used to measure the
boundary or coastline of an image. 1983 (5)
Tatsumi et al. 1919 (32)
Besicovitch. Smith.. 1983 (5)
Hausdorff. 1990 (6)
Jelinek and Fernandez.
and G. or yardstick dimension).. E. dilation
dimension
Calliper
DC
Box
counting
DB
Richardson dimension. T.
This method is based on counting the number of steps that give
a polygonal representation of an arbitrary object using different
calliper spans. (C) Dilation method. 35).. compass
dimension. Losa. 1983 (5)
Smith et al. Figures 3A and 4 show examples of this method. 1990 (16)
Smith et al. R. Reprinted. 1989 (45)
Smith et al.
divider... F.). See text for
more details. Birhauser. 1990 (16)
Peitgen et al.. mosaic
amalgamation dimension.
Calliper Method
FIG.
That is. Eds. 1961
Mandelbrot. 1983 (5)
Takayasu. Lange (1998) in Fractals in Biology and Medicine (Nonnenmacher.
but it cannot be strictly applied
to natural objects due to its finite
range of fractal structure
Easier to evaluate than DH. 1989 (10)
Takayasu. divider
dimension..
An algorithm based on the Hausdorff dimension
is the calliper dimension (also known as the compass. 3. The length of the
coastline then equals the size of the ruler times the
number of steps r has taken to trace the coast. with various
diameters and centered on the border of the Koch island. 1991 (29)
Richardson. 1992 (48)
Caserta et al. from T. The capacity
dimension is related to the box counting and mass–
radius methods that are its applied. Basel. To determine D.. One
finds that the boundary length is a function of the
span of the calliper employed in the measurement. (A) Calliper method. Note loss of
border detail shown in (A) and (B) (D) Mass method example
after application of six groups of concentric disks. 1995 (7)
Caserta et al.
. D. the length does not converge to a stable value
but keeps increasing as the calliper span decreases. The difference from the Hausdorff–
Besicovitch dimension is that the set is now covered
with spheres of identical radius (16). All
centers lie within the radius of gyration (large circle). perimeter
dimension
Capacity dimension.
D(0) in multifractal analysis
Mass
DMR
Hausdorff–Besicovitch
dimension
Mass fractal dimension. The “capacity
dimension” has become the fundamental definition of
fractal dimension in the minds of many. G. two-dimensional
embodiment and described below.(B) Box counting method. mass
radius dimension. 1983 (5)
Mandelbrot. A. G.
1998 (59)
very difficult and a more practical parameter of D. 1989 (10)
Schroeder. 1989 (10)
Mandelbrot. box
dimension. was introduced by Kolmogorov (34. with permission. Jr. 1935 (33)
Mandelbrot. Kolmogorov
dimension. can also be applied
to surfaces and biological
structures
Reference
Mandelbrot.´ NDEZ AND JELINEK
FERNA
312
TABLE 1
Some of the Most Widely Used Fractal Dimensions with their Synonyms and Contexts
Dimension
Symbol
Synonyms
Fractal
Hausdorff
D
DH
Minkowski–
Bouligand
DM
Minkowsky sausage
dimension..
the capacity dimension. Some methods used for determination of fractal dimensions of a Koch triadic island with a D ⫽ 1.

(A) Measuring the length of the coastline of the Australian
continent. (B) Graph of resulting log–log plot.USE OF FRACTAL THEORY IN NEUROSCIENCE
313
FIG.
. Calliper method for ascertaing the boundary length of an image. 4.

The following methods can be used for noncontiguous structures as well as for 2D and 3D images.
6B.html. as reported by
the above articles. A common form of
this algorithm. The method is
illustrated in Fig. noncontiguous structures
or closed loops within a structure). 3C. 7). The
number N(r) of pixels constituting the image is
counted and at the same time the scaling factor r
of pixels is recorded. A Macintosh program for calculating D
using this method can be found at the following
URL: http://plantecohost. S. called
the Minkowski “sausage. The length of the border for each respective
diameter is determined by the area of the outline
divided by the diameter. (10). The box counting method applies to
any structure in the plane and can be adapted for
structures in three-dimensional space (7. the result will be the
length of the curve. For a fractal curve the length
will continue to increase as the radius of the circles decreases. The
sequence of box sizes for grids is usually reduced by
a factor of 1/2 from one grid to the next. The NIH Image program
and its many macros can be fetched in a number of
ways. the points
will fall on a straight line between an upper and
lower bound with negative slope. (5. A circle is swept continuously
along the line and the area that is covered.
47–49). In practice. The fractal dimension is
then estimated from the slope of the log–log plot of
length against diameter. 36.314
´ NDEZ AND JELINEK
FERNA
If the length of the boundary (coastline) versus the
calliper length is plotted on a log–log scale. as devised by Flook (50).” is determined. This is done by application of a convolution
procedure which is part of the image analysis program (dilation macro from NIH). the image is digitized by
pixels having a given scaling factor r (Fig.harvard. with
the initial box size being the size of the image. A macro for this method can be obtained from the National Institutes of Health (NIH)
at ftp://codon. the Euclidean space containing the
image is divided into a grid of boxes of size r. Measurement of N(r) at larger
scaling factor (lower resolutions) is usually done by
zooming down the image using the memory frame
with four adjacent pixels making one pixel (Figs.g. is counted (Fig. The pixel dilation method.nih. D is then calculated by fitting a
linear regression to the following equation: log(r)
⫽ ⫺D* log(n) ⫹ K. where r ⫽ resolution of image
(number of pixels per unit length)..info. 43. These are detailed at http://rsb. The grid intercept method relies on progressively coarsening the image representation (by
pixels having different scaling factors) and counting the number of pixels intersecting a portion of
the image (Fig.
Many research reports using this scheme to analyze neuron structures are found in the literature
(10. This value
is then plotted as a function of the circle diameter.
Pixel Dilation Method
The pixel dilation method is based on the Minkowski–Bouligand dimension. is that images composed of more than one simple perimeter cannot be
processed accurately (e. (10) and others (11. 36).gov/pub/nih-image/user-macros/
box count macro.
A method similar to the box counting technique
is the grid intercept method (45. 46).
51–56). or
gradient is related to the fractal dimension by D ⫽
1 ⫺ S (5).gov/
. for use with NIH Image image processing software. Note that the
terms box counting method and grid intercept
method refer generally to two different methods
but are used interchangeably in the literature (16. Euclidean curve. r is
then made progressively smaller and the corresponding number of nonempty boxes. 6). Figure 5
shows an example of this calculation for a retinal
ganglion cell. This filters out
structures smaller than the current diameter of the
circle. 37–42). The calliper method has previously been
used to characterize neurons (10. and K
Minkowski–Bouligand Dimension
The Minkowski–Bouligand dimension is different
from the Hausdorff dimension (18).
3B).
and the slope (on the usual log–log plot) gives the
dimension.
Box Counting Method
To estimate D.nih. The logarithm of N(r) versus the logarithm of r
gives a line whose gradient corresponds to D. One major
drawback of the calliper method. 12. has been
implemented by Smith et al. 6A). 44). N(r).
This is equivalent to the “grid” method described by
Smith et al. 10. n ⫽ the number
of pixels intersecting a portion of the image.txt. 29).edu/gmbWWW/
APPL.
is a constant. The slope. replaces each pixel of the border
by a circle whose diameter ranges from 3 to 61 pixels
(Fig. For
a smooth. The important difference between this
and the calliper method is that the circle is moved
so that its center lies on every point of the line. 6C).

FIG. Digitized images of a turtle ganglion cell. 5. Box counting method applied to a retinal ganglion cell (cat beta cell) demonstrates the halving of the box sizes of the grids
that overlay the image. (C) Scaling factor ⫽ 4 ␮m. Each image
frame has pixels with a different scaling factor. (A) Binary image of a turtle ganglion cell.
. 6. (A) Scaling factor
⫽ 1 ␮m. (B) Minkowski
“sausage” created by application of the dilation method to (A). (B) Scaling factor ⫽ 2 ␮m.
The perimeter is calculated as the area of this figure divided by
the diameter of the dilating disk.USE OF FRACTAL THEORY IN NEUROSCIENCE
315
FIG.
FIG. 7.

If one takes
a fraction of the radius of gyration. Usually the
quantity of interest is the area of the image.edu.gov/pub/
nih-image/user-macros/. This can happen with a known fractal
such as a Koch snowflake. when most
of the mass is concentrated in a convex outer border
the method totally fails because the radius of gyration falls tightly within the border itself. one could claim that it is only the border
that is fractal. This premise stems from the
fact that the computer screen has a limited resolution
and may not be able to represent branching patterns
below the size of one pixel. all appropriate fractal analysis methods approach the same limit.
For strictly self-similar mathematical fractals
such as the Koch curve. There are various versions for various Macintosh computers. To lessen
computation time a fraction. However. M. monomers in a
polymer chain. The method first computes the center of
gravity and then the radius of gyration. named Fractop. This particular macro (fractal dilation. the cell body and/or the axon may also be removed from binary or outline representations of
neurons.
IMAGE PREPARATION AND DETERMINATION
OF D
Digitized images can be presented as binary. on the other hand. or border-only images of
cat retinal ganglion cells as long as the dendrites are
thin with respect to the cell body. The double logarithmic plot
of M(r) against r gives a quantitative value for D. Therefore when calculating the D using
complete binary images of neurons there may be a
space-filling effect that can lead to a higher D or a
D of 2. However.au/
fractop/ and discussed by Jones and Jelinek in this
issue. Note that it is necessary to sample all local
origins to sample as many data points belonging to
the image as possible. Previous results (58)
have demonstrated no significant difference between
the estimated D of binary images. border
roughness (Fig. 8). adsorption sites on a surface. For instance.nih. This is due to
the area of the cell body and dendrites being much
smaller than the extent of the border (58.
One is related to how image presentation may influence the possible scaling relationship of the image
and the associated estimated D. This version. there are two further considerations.txt) and other user contributed macros are found at ftp://codon. the entire border
falls outside. etc. can be used and every point within this
limit is then chosen as a local origin and the cluster
mass (number of pixels occupied) within a distance
r of this local origin calculated. this finding is dependent on the type of cell
one analyzes and does not hold for glia cells (60). In
addition. The sites may be pixels obtained from box
counting. With neurons
specifically.
Mass–Radius Method
The mass–radius dimension is defined by the relationship between the sites of an image found within
a sphere or circle of a certain radius covering the
image.csu. the cell body interior and that of the
dendrites do fill a plane completely and hence have
a D of 2. with a D of
2 (60).
. a circle or sphere of radius r is laid over
the image. primary
particles of a colloidal aggregate.6. binary images with
cell body and axon removed.
Mandelbrot (5) stated that an object that fills a plane
completely has a dimension value of 2. 3D). had
significantly lower D values (58) since they represented only the dendritic branching and do not reflect
the other characteristic of complexity. Having decided which analysis
method to use. All possible choices
of local origin are averaged and the average cluster
mass M(r) is obtained. has the added
advantage of providing a choice for the number of
centers and the fraction of the radius of gyration
required. 57). The histological techniques used may also lead to incomplete staining of
the peripheral parts of the cell. depending on the relationship between the
internal area and the contour.316
´ NDEZ AND JELINEK
FERNA
nih-image/download. 59). Skeletonized images. skeletonized or border-only images. and the other is
related to estimating D from the data points. say 0. The radius of gyration is introduced as a method of avoiding the outer edges of the
figure based on the premise that the peripheral parts
of the image that represent natural objects such as
neurons is incomplete. A
multiplatform version for computing the mass fractal
dimension is available from http://life.
To implement this method for the analysis of 2D
images. the Hausdorff
dimension (29. the filled interior is solid.html. When analyzing neurons. steps of a random walk. The choice of format is related to the spacefilling attributes of the image and the attributes of
the image one deems to be important. of the radius
of gyration. that
increases with the increase in the radius r (see
Fig.

The actual data
points generally do not lie on a straight line for more
than one to two decades. the number of data points being related
to the number of measuring steps. indicating the
317
same image. some investigators have obtained linear
plots using skeletonized images of neurons. This limited self-similarity
or scale invariance is characteristic of biological material and is a focus of some controversy (51. as described above. obtained linear log–log plots (⬎ 2 generations)
with skeletonized images of retinal neurons. (6.
and skeletonized images (58). and skeletonized (C) images. outline-only. D is related to the slope of
the line. Analysis of skeletonized images using the original NIH Image box
counting method (Version 1.USE OF FRACTAL THEORY IN NEUROSCIENCE
GRAPHIC DETERMINATION OF THE
FRACTAL DIMENSION
How the actual D value is obtained from the log–
log data points can lead to differences in the magnitude of D. In such a plot. This dependency on the analysis method to produce linear log–log plots with skeletonized images may explain the conclusions of Panico
and Sterling (61). Montague and Friedlander (14). for instance. Box counting analysis of the same turtle ganglion cell. Caserta
et al. 7) using the mass–radius method. was not scale-invariant under this transformation and method (58). 8. also obtained linear log–log plots with
skeletonized images. The figures
on the bottom are the associated graphs of their fractal dimensions. using a different implementation of the box counting method
(greater number of box sizes) and different image
handling (rotation of image and using multiple centers). These authors used two variants
of the box counting method and the mass–radius
method with skeletonized images and concluded that
FIG. 61).
However. when skeletonized. using binary (A).
. outlined (B).2) led at times to a sigmoid log–log data point distribution.
Differences in the linearity of the log–log data
points was observed between binary.

07
(filled circles). however. of course.
use of the value with the longest linear range is suggested. which will
have an F distribution (49). The linear region can also be
calculated by determining the local slopes.41 (open circles) and 1.
be able to fit the data better. One
method for this. The simplest method
of obtaining D is to fit a regression line to all data
points and determine the slope of this line. however. and the improvement in the fit may not be that great. The range of linearity is not important if the D obtained in this way is used in differentiating between different cell types (Caserta.07. The
D with the smallest linear range (1. (7) for
the mass–radius method.318
´ NDEZ AND JELINEK
FERNA
cat retinal ganglion cells are not fractal due to their
limited linearity. The use of a hierarchical cluster
analysis to compute particular subsets of the log–log
values that achieve the best linear fittings (Fig.
In a recent study (12) we posed the following question: Can the estimate of D resolve differences in
neuronal branching when simpler metrical analysis
alone cannot? Our results indicated that although D
alone does not completely specify a cell’s morphology. Thus fractal analysis
.
When this method produces multiple values of D. as the difference in log N(r) divided by
log (r) for every n successive points.
The higher-degree polynomial will always. This method. 9)
has also been reported (12). personal communication). The left-hand side shows the digitized image of a retinal
bipolar cell. The test is performed
using a critical value of F ( p ⫽ 0. The region in
which the local slopes are constant is then taken as
the linear region (7).41. it is a statistically significant parameter for identifying and differentiating neuronal cell classes. 53.25). but it uses up one more
degree of freedom in the process. This technique allows
the detection of changes in D at different scales of
measurement and compensates for the finite size effects induced by the limited resolution of the images. Deciding on the range of
linearity and especially if it is significance has been
addressed by Russ (49). other methods included only
points that fell on the straight part of the line and
excluded data points obtained from the peripheral
parts of the image (41).
Panico and Sterling (61) also used the local slope
method to determine the D of their images. 61).
from a statistical point of view such a method would
not be justified. is to calculate the n-point
local slopes.
ADVANTAGES AND POTENTIAL PROBLEMS
OF FRACTAL DIMENSIONS
FIG. 52. The right-hand side shows a plot of the box counting
measurements.
and indeed it should not be expected to.
Because of the limited scale invariance of neurons
different authors have used different methods to determine D from log–log values. A hierarchical cluster analysis yielded two regression lines with two different D values: 1. described by Caserta et al. Their
conclusion was that the region of true linearity of
the local slopes was less than one generation and
therefore the images analyzed were not self-similar
and could not be fractal. The test is
based on the ratio of reduced ␹2 values. filled circles) could be
attributed to finite size effects at very low scales. the questions of whether an image
is fractal and whether an image belongs to a certain
group based on the D value are different and need
to be disentangled. who suggested that comparing the fit of the data points to a straight line and
to a higher-degree polynomial can clarify whether a
straight-line fit is an appropriate model of the data. Several publications have used this
method to determine the slope of the data points
(10. Alternatively. One of the main ones is that the sensitivity changes as the window over which the local
slopes is obtained is decreased (62). has
several flaws. 37. Method for the graphic determination of fractal dimensions.
Clearly then. 9. Therefore the
linearity region increases as the window is increased
and makes this is a very subjective method. If the linear
fit is accepted then the image is fractal. as biological objects display statistical self-similarity only between a short range of dimensions. This method considers the D of the cell drawing to
be the one with the longest linear range (1. open circles).

higher D
values are obtained by using the mass fractal methods than by using the pixel dilation and box counting
procedures. To capture all this richness of
this complex structure into a theoretical model is one
of the major challenges of modern theoretical biology
(64). summarizes concisely and meaningfully the amount of detail. Not even the “coastline of Britain” example
in Mandelbrot’s seminal work (5) has a power law
behavior spanning more that one or two orders of
magnitude (69). 64).
319
Furthermore whether a higher fractal dimension
would correlate with a more complex physiological
response is still an unresolved issue (9. 62).
Although all analysis methods rely on the relationship between a measuring device and the object’s
spatial distribution. Many neurons display
irregular shapes and discontinuous morphogenetic
patterns in support and in connection with their
functional diversity. This contrasts with integer-dimensional
measurement of anisotropic objects which require
multiple samples through the thickness of the threedimensional objects (1). 19). This means that D values
of specimens that have been processed in different
batches or at different laboratories can usually be
compared directly (as long as the same methodology
to calculate fractal dimension is used). space filling. the connection
between empirical values of D and any specific
growth mechanism should be avoided and require
the answering of further experimental questions. such us dendritic field extent and total
dendritic length. to more complicated global.
Furthermore fractal geometry has some other advantages over its integer-dimensional counterparts.
CONCLUSIONS AND FUTURE
DEVELOPMENTS
Fractal analysis has already found widespread application in the field of neuroscience and is being
used in many other areas.
These parameters range from simple metrical descriptors.
A further advantage of fractal analysis is that
shrinkage or expansion of a specimen will not affect
D as long as the artifact acts equally in all directions
and the measured points still lie on the linear segment of the graph (19). however.
39. It has thus become important to establish some criteria for choosing a particular method
and how these methods compare in order to standardize the computation of D (59). 12. not all methods give identical
results for the same form. and various authors have discussed classification systems of neurons using fractal analysis (7. Finally. be kept in mind that D is only
a descriptive parameter. Notwithstanding the above-mentioned limitations. Thus. These data reinforce the idea
that comparison of measurements of different profiles using the same measurement method may be
useful and valid even if the exact numeric value of
the dimension is not necessarily very accurate. 12). some
of the images analyzed using fractal analysis may
not demonstrate self-similarity or scale invariance
over more than one or two levels of magnification
and may not be fractal (61. Thus many quantitative parameters have been
used to characterize the morphology of nerve cells.
Thus in almost all circumstances the fractional component of dimension is retained when a fractal object
is projected to a lower-order dimension (18. the
fractal dimension. and
does not necessarily imply any underlying mechanism of form generation. the results
are always consistent. it remains that in many situations a single number. an
example being the projection of three-dimensional
retinal ganglion cells onto a two-dimensional film or
drawing (7). 63). Furthermore whether
scale invariance is observed for a particular image
is dependent on image presentation and the analysis
program applied to obtain the final D (59. Our results using
different methods to compute the D values show that
although different measurement procedures and
even the same algorithm performed by different computer programs and/or experimenters may give
slightly different numerical values of D. In general.
Unlike mathematically generated fractals. Thus determining D of a neuron.
A basic consideration is that most measurements
cover only a relatively short range of dimensions. like the dendritic field area
or the number of segments of a dendritic tree. that can be used for an objective
assessment of the degree of complexity (a concept
heretofore not readily quantifiable) of developing and
mature neurons. could immensely aid in the morphological discernment of different neuron types or neurons that
. 42. biological data that have a linear fit of more
than two orders of magnitude are extremely rare
(66–69). For example. 62.
It should. or complexity of neurons. 59. in
addition to the other morphometric criteria typically
used.USE OF FRACTAL THEORY IN NEUROSCIENCE
has an important role in characterizing natural objects. real data
cannot be ideally fractal over all scales. descriptors such as D.